Abstract

The formation of multiplexed phase-only holograms with more weighted phase functions creates spurious cross terms and nonlinear scaling. We extend previously reported work [Appl. Opt. 25, 3767 (1986)] by proposing a normal method to analyze multiplexed holograms mathematically. We show that the output of holograms with any number weighted phase function can be written as a new linear combination for the original phase function with new weights. The relationship between the original weights and the new weights is developed for real-time optimization of hologram performance. We focus on the analysis of two and three multiplexed holograms to demonstrate the effectiveness of this approach.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. D. P. Casasent, �??Unified synthetic discriminant function computational formulation,�?? Appl. Opt. 23, 1620�??1627 (1984).
    [CrossRef] [PubMed]
  2. J. A. Davis, S. W. Connely, G. W. Bach, R. A. Lilly, and D. M. Cottrell, �??Programmable optical interconnections with large fun-out capability using the magneto-optic spatial light modulator,�?? Opt. Lett. 14, 102�??104 (1989).
    [CrossRef] [PubMed]
  3. J. A. Davis, D. M. Cottrell, R. A. Lilly, and S. W. Connely, �??Multiplexed phase-encoded lenses written on spatial light modulator,�?? Opt. Lett. 14, 420�??422 (1989).
    [CrossRef] [PubMed]
  4. S. Reichelt and H. J. Tiziani, �??Twin-CGHs for absolute calibration in wavefront testing interferometry,�?? Opt. Commun. 220, 23�??32 (2003).
    [CrossRef]
  5. M. Beyerlein, N. Lindlein, and J. Schwider, �??Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics,�?? Appl. Opt. 41, 2440�??2447 (2002).
    [CrossRef] [PubMed]
  6. J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, �??Effects of sampling and binarization in the output of the joint Fourier transform correlator,�?? Opt. Eng. 29, 1094�??1100 (1990).
    [CrossRef]
  7. J. L. Horner and P. D. Gianino, �??Applying the phase-only filter concept to the synthetic discriminant function correlation filter,�?? Appl. Opt. 24, 851�??855 (1985).
    [CrossRef] [PubMed]
  8. D. P. Casasent and W. A. Rozzi, �??Computer-generated and phase-only synthetic discriminant function filters,�?? Appl. Opt. 25, 3767�??3772 (1986).
    [CrossRef] [PubMed]
  9. R. R Kallman, �??Optimal low noise phase-only and binary phase-only optical correlation filters for threshold detectors,�?? Appl. Opt. 25, 4216�??4217 (1986).
    [CrossRef] [PubMed]
  10. E. Carcole, M. S. Millan, and J. Campos, �??Derivation of weighting coefficients for multiplexed phase-diffractive elements,�?? Opt. Lett. 20, 2360�??2362 (1995).
    [CrossRef] [PubMed]

Appl. Opt.

Opt. Commun.

S. Reichelt and H. J. Tiziani, �??Twin-CGHs for absolute calibration in wavefront testing interferometry,�?? Opt. Commun. 220, 23�??32 (2003).
[CrossRef]

Opt. Eng.

J. A. Davis, E. A. Merrill, D. M. Cottrell, and R. M. Bunch, �??Effects of sampling and binarization in the output of the joint Fourier transform correlator,�?? Opt. Eng. 29, 1094�??1100 (1990).
[CrossRef]

Opt. Lett.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Input ratio versus the desired output ratio of the three multiplexed holograms.

Fig. 2.
Fig. 2.

Diffraction efficiency versus output ratio of the three multiplexed holograms.

Fig. 3
Fig. 3

(a) Input ratio and (b) diffraction efficiency versus the desired output ratio of two multiplexed holograms.

Fig. 4.
Fig. 4.

(a) Binary representation of a trifocal lens desired for x 1 = 0.5 and x 2 = 0.6 and (b) reconstruction.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

M exp ( ia ) = n = 1 N A n exp ( i ϕ n )
exp ( ia ) = n = 1 N A n M exp ( i ϕ n )
M = [ A 1 2 + A 2 2 + + A N 2 + 2 A 1 A 2 cos ( ϕ 1 ϕ 2 ) + 2 A 1 A 3 cos ( ϕ 1 ϕ 3 ) + + 2 A 1 A N cos ( ϕ 1 ϕ N )
+ 2 A 2 A 3 cos ( ( ϕ 1 ϕ 3 ) ( ϕ 1 ϕ 2 ) ) + + 2 A 2 A N cos ( ( ϕ 1 ϕ N ) ( ϕ 1 ϕ 2 ) )
+ ⋯⋯
+ 2 A N 1 A N cos ( ( ϕ 1 ϕ N ) ( ϕ 1 ϕ N 1 ) ) ] 1 2
M ( β 1 β 2 β N 1 ) = 1 M = m 1 m N 1 a m 1 m 2 m N 1 exp ( i m 1 β 1 + i m 2 β 2 + + i m N 1 β N 1 )
β 1 = ϕ 1 ϕ 2 β N 1 = ϕ 1 ϕ N 1
a m 1 m 2 m N 1 = 1 ( 2 π ) N 1 0 2 π 0 2 π M ( i m 1 β 1 i m N 1 β N 1 ) d β 1 N 1
exp ( ia ) = m 1 m N 1 a m 1 m N 1 { A 1 exp [ i ( m 1 + + m N 1 + 1 ) ϕ 1 i m 1 ϕ 2 i m N 1 ϕ N ] + A 2 exp [ i ( m 1 + + m N 1 ) ϕ 1 i ( m 1 1 ) ϕ 2 i m N 1 ϕ N ] + + A N exp [ i ( m 1 + + m N 1 ) ϕ 1 i m 1 ϕ 2 i £ m N 1 1 £ © ϕ N ] }
exp ( ia ) = + ( a 00 A 1 + a 10 A 2 + a 01 A 3 ) exp ( i ϕ 1 ) + ( a 00 A 2 + a 10 A 1 + a 11 A 3 ) exp ( i ϕ 2 )
+ ( a 00 A 3 + a 0 1 A 1 + a 1 1 A 2 ) exp ( i ϕ 3 ) +
y 2 = i = 0 k j = 0 i m ij x 1 j x 2 i j , y 1 = i = 0 k j = 0 i n ij x 1 j x 2 i j
y 1 = 0.00005096072781 + 3.94783 x 1 0 . 00873 x 2 10.19480 x 1 2 + ⋯⋯ + 11.96649 x 1 2 x 2 5 +
4.82817 x 1 x 2 6 4.11001 x 2 7
y 2 = 0.00005096072782 0.00873 x 1 + 3.94783 x 2 + 0.44625 x 1 2 + ⋯⋯ + 8.09256 x 1 2 x 2 5 +
6.93143 x 1 x 2 6 + 2.49845 x 2 7
exp ( ia ) = m 1 m 2 a m 1 m 2 { A 1 exp { ik 2 f [ [ x + ( m 1 a m 2 b ) ] 2 + y 2 ] } exp [ ik 2 f ( ( m 1 a m 2 b ) 2 m 1 a 2 m 2 b 2 ) ] + A 2 exp { ik 2 f [ [ x + ( m 1 a m 2 b a ) ] 2 + y 2 ] } exp [ ik 2 f ( ( m 1 a m 2 b a ) 2 m 1 a 2 m 2 b 2 + a 2 ) ] + A 3 exp { ik 2 f [ [ x + ( m 1 a m 2 b + b ) ] 2 + y 2 ] } exp [ ik 2 f ( ( m 1 a m 2 b + b ) 2 m 1 a 2 m 2 b 2 + b 2 ) ] }

Metrics